Multiport feedback and polezero control

ABSTRACT

Apparatus and methods are disclosed which allow the critical frequencies (poles and zeros) to be independently controlled by applying state-variable feedback techniques to both the external and internal ports of an arbitrary linear, passive, and timeinvariant network (system). There are no limitations on the locations of either the poles or zeros and the invention allows the generation of RC, RL, LC, and RLC and nonpositive real driving-point impedances and transfer functions from active RC networks. Therefore, if one fabricated any transfer function with these active driving-point impedances, pole-zero control will be achieved for the transfer function.

United States Patent DAlessandro [54] MULTIPORT FEEDBACK AND POLE- ZERO CONTROL [72] Inventor: John R. DAlessandro, 2970 North Sheridan Road, Apt. 1426, Chicago, Ill. 60657 [22] Filed: Nov. 17, 1969 [21] App]. No.: 877,402

[52] U.S. Cl. ..333/80 R, 330/l09 [58] Field ofSearch ..333/80,80T,85, 107; 330/ 109 [56] References Cited UNITED STATES PATENTS 2,823,357 2/1958 Hall, Jr ..333/80 T 2,924,781 2/1960 Wilson et al. ..330/85 X 1 Feb.ll5,ll972 FOREIGN PATENTS OR APPLICATIONS 548,544 11/1957 Canada ..330/l09 Primary Examiner-Herman Karl Saalbach Assistant Examiner-Paul LI Gensler Attorney-Hill, Sherman, Meroni, Gross & Simpson [57] ABSTRACT Apparatus and methods are disclosed which allow the critical frequencies (poles and zeros) to be independently controlled by applying state-variable feedback techniques to both the external and internal ports of an arbitrary linear, passive, and time-invariant network (system). There are no limitations on the locations of either the poles or zeros and the invention allows the generation of RC, RL, LC, and RLC and nonpositive real driving-point impedances and transfer functions from active RC networks. Therefore, if one fabricated any transfer function with these active driving-point impedances, pole-zero control will be achieved for the transfer function.

10 Claims, 20 Drawing Figures RC PAS SIVE NETWORK PATENTEDFEB I5 I972 BIENBI l 8N SHEET I OF I IO 15 B F E 2;

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L a: L L V v n K E0 SUMMING BUFFER I R 3 CL AMPLIFIER AMPLIFlER F G I 9 INVENTOR JOHN DALESSAN'DRO PATENYEDFEB 15 I972 SHEET Q 0F 4 BY we 1 i (usually) SUMWNG V AMPLIFIER AM AMPL\F\ ERS I F] G. 20 K'= l INVENTUR.

A TTORNE YS DIFFERENTlAL MULTIIPOIRT FEEDBACK AND POLE-ZERO CON'II'RGI.

BACKGROUND OF THE INVENTION l. Field ofthe Invention This invention relates in general to controlling the response of networks or other systems and in particular to a system for controlling the critical frequencies (poles and zeros) of an arbitrary driving-point immittance.

2. Description of the Prior Art As integrated circuits have come more and more in use in electronics, it has been desirable to synthesize inductive reactance because capacitors and resistors are much more easily formed on substrates and printed circuit boards than are inductances. It has been common to use hybrid circuits wherein inductors wound on suitable coil forms are mounted on a print circuit board and have leads that attach to the print circuit of the board.

SUMMARY OF THE INVENTION The invention described herein was made in the course of or under a grant from the National Science Foundation, an agency of the United States Government.

The present invention allows the critical frequencies (poles and zeros) of an arbitrary driving point immittance to be controlled independently by applying state-variable feedback techniques to both the external and the internal ports of an arbitrary linear, passive and time-invariant network (system) without limitations on the locations of either the poles or the zeros. The invention allows inductive reactance to be synthesized from pure resistive and capacitive networks by utilizing feedback in the input terminal to control the zeros of the system and feedback at an internal port to control the poles of the system. Thus, if a designer of electronic components wishes to obtain a component with a desired function, the present invention allows networks to by synthesized for the particular function. If a circuit containing capacitors are util ized, the capacitive voltages may be measured and amplified and summed, and fed back to input and other ports of the network with feedback to synthesize the mathematical equation of the desired function. Thus, techniques are provided for synthesizing circuits of any desired response and in which the poles and zeros may be controlled as desired. Although the invention is primarily described with respect to electrical networks, it is to be realized that the techniques disclosed herein are applicable to many other problems and the response of a system such as hydraulic systems, heat systems, etc., may also be synthesized and controlled by utilizing the techniques of this invention.

Other objects, features, and advantages of the invention will be readily apparent from the following description of certain preferred embodiments thereof taken in conjunction with the accompanying drawings, although variations and modifications may be effected without departing from the spirit and scope of the novel concepts ofthe disclosure.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. I is a block diagram of a linear time-invariant one-port network;

FIG. 2 is the network of FIG. I which has been converted into a two-port network;

FIG. 3 is a block diagram with feedback to control the zeros;

FIG. 4 illustrates a particular configuration of the RC network of FIG. 3;

FIG. 5 is a schematic view of an RL circuit which can be synthesized with the circuit of FIG. 4;

FIG. 6 is a schematic view of a circuit that can by synthesized with the networks of FIGS. 3 and 4;

FIG. 7 is a schematic view of another RLC circuit that can be synthesized with the networks of FIGS. 3 and 4;

FIG. 8 illustrates another circuit that can be synthesized with the networks of FIGS. 3 and 4;

FIG. 9 illustrates another circuit that can be synthesized with the networks of FIGS. 3 and 4',

FIG. I0 is a block diagram with feedback in the input port and at any port other than the input port so as to obtain both pole and zero control;

FIG. 11 is a schematic view of a particular configuration of the circuit illustrated in FIG. I0 and comprises a first-order circuit;

FIG. 12 illustrates a second order circuit in schematic form of the circuit of FIG. I'll;

FIG. l3 illustrates a second order circuit of the form illustrated in FIG. 10;

FIG. I4 illustrates a second order circuit of RLC form with feedback to the input;

FIG. I5 is a block diagram for illustrating one of the theorems of this invention using admittance;

FIG. 16 is a schematic view for a circuit utilizing admittance control;

FIG. I7 is another schematic view ofa circuit utilizing control of admittance;

FIG. 18 is a block diagram illustrating a feedback voltage source according to the invention;

FIG. I9 is a block diagram of a feedback source and illustrates a buffer; and

FIG. 20 is an example of the implementation of the invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS I have discovered that independent pole-zero control can always be accomplished for any passive driving-point immittance by properly applying state-variable feedback techniques. In fact, it will be shown that pole control is dependent upon the application of state-variable feedback at some point(s) internal to the network. It should be emphasized that the following discussion also applies to RL, LC, and RLC passive networks, but will be presented in terms of RC networks since they are of particular interest with respect to the fabrication of integrated circuits. It should also be noted that the following network applies not only to networks, but more generally to the broader class of linear systems. In all cases, the state variables are defined as the capacitor voltages and the inductor currents, however, it is to be realized that other choices of state variables exist.

The State-Variable Approach A useful theorem developed by R. W. Brockett states that:

If a single-input, single-output, linear, time-invariant system is both controllable and observable, then linear state-variable feedback can be used to achieve any desired pole configuration consistent with the dimension of the system. Any number of the open-loop zeros can be cancelled, but no new zeros can be introduced. This theorem reveals that the poles ofa system (as defined by the characteristic polynomial, Ms), of the system) can be independently selected. by choosing an appropriate feedback arrangement to the input port, and I have discovered that similar control can be accomplished with respect to the zeros of the system. This allows one to manipulate a given system to obtain any desired pole-zero configuration consistent with the order of the system. That this is true may be proved as follows with reference to the figures.

Zero Control FIG. 1 illustrates a linear, time-invariant one-port network III that is both observable and controllable. Let this one-port network be converted into a two-port network by defining a second, but fictitious, port (l =0) taken across any two nodes, except the input nodes, of the network. This is illustrated by and the open-circuit voltage transfer function for this two port is given by:

Brocketts Theorem states that linear state-variable feedback can be applied to the input of a single-input, single-output system to achieve any desired pole configuration consistent with the dimension of the system. Since equation (2) describes such a system, control over the poles of G, (s) can be accomplished by an application of this technique. However, control over the poles of G implies control over the zeros of Z (s), the open-circuit (I =0) driving-point impedance measured at port one of FIG. 2. Now, when 1 0, as is required by equation (2), the configurations defined by FIGS. 1 and 2 are identical and Z (s) of the modified network (FIG. 2) defines the driving-point impedance, Z,,,(s), of the original network (FIG. 1). Thus, control over the zeros of Z,,,(s) is established. Since any network structure can be modified as defined by FIG. 2, state-variable feedback implies independent control over each of the zeros of any driving-point impedance that is associated with a linear, time-invariant system that is both controllable and observable. If FIG. 1 is described in matrix form by a set of n linearly independent loop equations defined A(s) M A (s) determinant Z (s) i( A cofactor where A(s) is the determinant of the loop impedance matrix, Z(s), and A (s) is the (ii)"' cofactor of Z(s). Thus, state-variable feedback applied to any port i(i=I 2, n) ofthe network implies control over the zeros of the driving-point impedance measured at that port. This fact, in turn, implies independent control over each of the zeros of the loop impedance determinant, A(s), by virtue of equation (4). Summarizing this result yields the following useful theorems:

Theorem 1 If A(s) is the determinant (characteristic polynomial) of the loop impedance (system) matrix, Z(s), of a linear, time-invariant system that is both controllable and observable, then linear state-variable feedback can be applied to achieve any desired zero configuration consistent with the degree of the polynomial, A(s).

Theorem 2 If Z(s)=P(s)/Q(s) describes the driving-point impedance of a linear, time-invariant network that is both controllable and observable, then linear state-variable feedback can be applied to achieve any desired zero configuration consistent with the degree of the numerator polynomial, P(s). Any number of poles can be canceled but no new poles can be introduced.

Theorem 2 is illustrated by FIG. 3, where the K are the feedback amplifier gain constants of amplifier l6 and the capacitor voltages, V,,(i=l, 2, n), describe the state variables.

To illustrate the variety of driving-point impedances which can be obtained from a given RC active network of the configuration illustrated by FIG. 3, consider FIG. 4 which has resistors R R2, and R and capacitors C and C connected as shown. For computational convenience, the element values have been defined as illustrated in FIG. 4. The driving-point impedance of this structure is:

Example 1 If K,==2 and K =l 1/4, then equation (5) can be written as:

Z (S)=Z(S, 2, lI/4)=(S -l-2S+3/4)/(S -l4S+3) (6) As can be verified, this is the driving-point impedance descrip' tion of the RL network illustrated in FIG. 5. Note that an infinite set of values for K, and K exists for equation (5) to describe an RL network. The only constraint on the RL net works that can be obtained from the configuration of FIG. 4 is the location of the poles as described by the denominator polynomial of equation (5). By selecting an appropriate set of element values in the configuration of FIG. 4, any desired pole configuration consistent with an RC network can be achieved. However, the feedback technique described above will not permit control over the pole positions once the element values have been selected.

Example 2 To illustrate the variety of RLC and nonpositive real networks that can be obtained, consider the following network functions which can be realized by an appropriate selection of values for K and K in equation (5 The network realizations of equation (7), equation (8), equation (9), and equation (10) are illustrated in FIGS. 6, 7, 8, and 9 respectively.

For one-port networks, it has been shown that control can be exercised over the zeros of a driving-point impedance by the application of state-variable feedback techniques. This control is obtained by an appropriate selection of the feedback amplifier gains, K,. It should be noted that no control can be exercised over poles of a driving-point impedance. This lack of pole control is the only constraint on the class of active driving-point impedances which can be realized from a given passive structure for the configuration illustrated by FIG. 3, In the following section, this constraint will be removed.

Pole Control Theorem 1 will prove useful in establishing an important property associated with A,,(s), the (i, i cofactor of the loop impedance matrix, Z(s) A (s) can be interpreted in the following manner: it describes the determinant, A(s), of the loop impedance matrix, Z(s), obtained from Z(s) by deleting row i and column i of Z(s). Physically, this can be interpreted as open-circuiting porti (I i=0) of the original network. If we now apply state-variable feedback in series with any remaining port (chord) of the modified network as defined by Z'(s) an application of theorem 1 reveals that each of the zeros of A '(s)=A (s) can be controlled independently. Summarizing this result yields:

Theorem 3 If A (s) is the (i, i)"' cofactor of the loop impedance (system) matrix, Z(s) of a linear, time-invariant system that is both controllable and observable, then linear state-variable feedback can be applied internally to achieve any desired zero configuration consistent with the degree of the polynomial, id-

Theorem 3 together with equation (4) establishes that linear state-variable feedback can be applied internally to a network and will result in independent pole control for the driving-point impedance of the network. Thus, the following theorem can be stated:

Theorem 4 If Z(sFP(s)/Q(s) describes the driving-point impedance of a linear, time-invariant network that is both controllable and observable, then linear state-variable feedback can be applied internally to the network to achieve any desired pole configu-' ration consistent with the degree of the denominator polynomial, Q(s).

Pole-Zero Control Theorem 2 and theorem 4 or theorem 1 and theorem 3 together with equation (4) establish that each of the poles and the zeros of a driving-point impedance can be controlled independently. Internal feedback determines the poles of the impedance. Once these have been defined, an application of state-variable feedback to the input port then determines the zeros of the driving-point impedance. This order of selecting the critical frequencies of a driving-point impedance is advantageous since the feedback factors, say K j l 2, n, associated with the internal feedback appear in both the denominator and numerator polynomials of the driving-point impedance. The feedback factors, say K l, 2, n, associated with the input port appear only in the numerator polynomial of the driving-point impedance function.

Pole-zero control is illustrated by FIG. It) with RC network having feedback amplifiers 22 and 23.

Theorem 5 If Z(s)=P(s)/Q(s) describes the driving-point impedance of a linear, time-invariant network that is both controllable and observable, then linear state-variable feedback can be applied to achieve any desired pole-zero configuration consistent with the degrees of the polynomials, P(s) and Q(s). Feedback applied internally to the network determines the poles of the impedance. The zeros of the impedance are then determined by applying state-variable feedback to the input port.

It should be emphasized that the internal feedback can be applied to any port as shown in FIG. other than the input port of the network. In order to illustrate the multiport feedback technique, several examples will be given.

Example 3. First Order System Consider the two-port network illustrated by FIG. 11. The driving-point impedance, Z,,, (s), measured at port two with port one short-circuited is described by with C- =C 1+ and C =C /(l+ Thus, independent pole-zero control exists.

A calculation of the open circuit voltage transfer function yields:

Note that both the pole and the zero of this function can be controlled independently by the feedback factors, K and K I believe as illustrated by the following, that multiport feedback can accomplish independent pole-zero control for the open-circuit voltage transfer function between any two ports of an n-port passive network to which state-variable feedback has been applied. As shall be demonstrated in the following examples, this is indeed the case.

Example 4. Second order System A more complex structure than that which was studied in the previous example is illustrated by FIG. 12.

A calculation of the driving-point impedance, Z,,, (s), measured at port two with port one short-circuited yields For this function, the factors, K and K define the poles which can be made to be real or complex conjugates. The factors, K and K define the zero and amplitude of equation (13).

The open-circuit voltage transfer function for this system is described by Another example of a multiport feedback system is illustrated in FIG. 13. In order to reduce the algebraic complexity involved in describing this system, all passive elements have been arbitrarily assigned a value of unity. The open-circuit voltage transfer function between port one and port two of this structure is described by Except for the coefficients ofthe S terms, each ofthe coefficients of equation 15) can be selected independently of the others. As a result, any combination of second order poles or zeros, either real or complex, can be obtained. Since the feedback factors, K can only assume real values, any complex poles or or zeros must each occur in complex conjugate pairs.

It should be noted that the the three systems described above by equation l2), equation l4) and equation (15) are sufficient to generate any open-circuit voltage transfer function whose numerator and denominator polynomials are each of an arbitrary degree and such that the degree of the numerator polynomial is less than or equal to the degree of the denominator polynomial.

Although the various examples given in this section are RC structures, the preceding discussion and theorems also apply to RL, LC, and RLC networks. In all cases, the state variables are defined as the capacitor voltages and the inductor currents. For example, the driving-point impedance of the network illustrated by FIG. 14 is given by:

in (S, ISL; (LGCIB) m-HQ) SHIP Theorem5 shows that each of the poles and zeros of any driving-point impedance can be controlled independently by state-variable feedback techniques. Therefore, if one fabricated any transfer function with these active drivingpoint impedances, pole-zero control will be achieved for the transfer function.

One important advantage of the technique described above is that pole-zero control can be exercised at any time simply by altering the feedback gain constants, K This capability is particularly attractive for use with integrated circuits, since the passive elements can be fabricated as one module and the active elements as another and then interconnected. Each active module would then determine a particular pole-zero configuration. It would then be a matter of interchanging active and/or passive modules to achieve any desired pole-zero configuration.

It should be noted that although various networks on a loop impedance basis have been described, other network characterizations are also possible. For example, on an admittance basis, the feedback sources could be represented as dependent current sources and would be applied in parallel with each input (current) source at each port of the network. This is illustrated by FIG. 15 for an RC l-port network. By proceeding in a manner similar to that involving the loop impedance matrix, Z(s), analogous results to those illustrated previously, but expressed in terms of the admittance parameters of a network, can be obtained. For example, the driving-point admittance of the network illustrated by FIG. 16 is given by equation (17):

As was expected, pole-zero control is achieved for this fu nction. The internal feedback factor, K determines the location of the pole of this function, and the input feedback factor, K,, then determines the zero of this function. Thus, theorem 5 also applies to admittances when the structure illustrated by FIG. 15 is used. The proof of this statement is identical to that given for theorem 5, except that the derivation is carried out on an admittance basis.

Theorem 6 If Y(s)=P(s)/Q(s) describes the driving-point admittance of a linear, time-invariant network that is both controllable and observable, then linear state-variable feedback can be applied to achieve any desired pole-zero configuration consistent with the degrees of the polynomials, P(s) and Q(s). Feedback applied internally to the network determines the poles of the admittance. The zeros of the admittance are then determined by applying state-variable feedback to the input port.

Another interesting feedback possibility is that of feeding back the derivatives of the state variables (i.e., capacitor currents and inductor voltages). Thus, for RC networks, the feedback sources could be made to be current-controlled current sources (transistors). Since transistors are easy to fabricate in integrated form, this technique should prove quite useful in integrated circuit design applications. For example, the drivingpoint admittance of the network illustrated by FIG. 17 is described by equation (18 Although several techniques might be applied to form the actual feedback sources, the following technique is suggested. The feedback voltage sources which are labeled consist of a summing amplifier preceded by a set of N differential amplifiers, K,, at each input to the summer. The details of this block are shown in FIG. 18. The double-arrow into this block represents the various capacitor voltages which are being summed by the summing amplifier. In FIG. 19, the block following the summing amplifier is simply a unity gain voltage amplifier whose input impedance is assumed to be infinite and whose output impedance is resistive and of value, R This configuration was chosen so that the current, I, through the passive RC network is not significantly altered because of the voltage feedback arrangement. The assumptions are reasonable and can be met in practice. If the amplifiers do have a finite input impedance, then the expression for the driving-point impedance will change; however, the control over the poles and zeros of this impedance will still be maintained. It should also be noted that if all capacitors are grounded then one can dispense with the differential amplifiers and feed each of the capacitive voltages directly into the operational (summing) amplifier. For simplicity, the impedances Z, and Z,could be resistive.

Thus, I have discovered that zero control may be accom plished by summing the capacitive voltages (for systems such as shown in FIGS. 3 and 4') and feeding this signal into the input port. As shown in FIG. 18 the capacitive voltage V ls supplied to an amplifier 50 and multiplied by K, to obtain K,V In a like manner the voltageV would be supplied to amplifier and multiplied by K to obtain K Y The outputs of these amplifiers would be summed to obtain I(,V plus K V which is the signal supplied by amplifier 16 in FIG. 4 to the input port of the circuit. In a similar manner the poles will be controlled by detecting the state-variable voltages, summing them with suitable coefficients and feeding the combined signal back at a port other than the input port to control the poles.

For a concrete example of the utilization of this invention to obtain pole and zero control, assume the problem wherein a design desires a circuit which has a desired function and a network must be synthesized for this function. For example, assume that the function of equation (6) is desired and assume the corresponding network illustrated in FIG. 5. By applying suitable gain constants to the circuit of FIG. 4 the configuration of the RC network of FIG. 4 will respond in the same manner as the LR network of FIG. 5. The gain constants may be obtained by equating the numerator polynomial coefficients to solve the gain constants. To solve for the coefficients of the numerator, corresponding coefficients of equations (5) and (6) are equated to yield a solution for the gain constants K,. Particular coefficients of the 5' terms can always be reduced by dividing or multiplying. Therefore, the gain constants can be obtained to achieve the desired network. Therefore, an active RC circuit is produced which synthesizes the desired RL circuit. It is also interesting to note that circuits may be synthesized which have negative elements (not normally realized) for example, see FIG. 9. It is to be realized that FIG. 4 is only a special case of the general configuration of the figures. The more capacitors the more zeros.

For pole control the designer is given the function and told to synthesize that function. Only the gains of an operational amplifier need be changed and those gains are determined by the ratios of impedances (thus a uniform variation in impedance values does not affect the circuit performance: for example, KR IKR RJR because K cancels out). By similar techniques as illustrated, for example, in FIG. 10 by summing the amplifier applied capacitor voltages and feeding these back in series within a chord (any other loop except input loop) allows the poles to be selected. It is generally desirable to first solve the pole pattern to obtain the internal feedback gain, and then solve the zero pattern to obtain the input amplifier constants. Thus, it is seen that this invention provides means for synthesizing the desired networks and utilizes feedback techniques to obtain the desired response.

FIG. 20 illustrates one technique that might be employed to feedback the amplified capacitive voltages to the input of the network. A similar technique could be applied to feedback these voltages at any other point in the network. Initially the capacitor voltages are amplified by a set of differential amplifiers'60 and 61. The outputs of these differential amplifiers are then fed into a summing amplifier 62 (an operational amplifier). The output of the summing amplifier is then fed into a buffer amplifier 63. The configuration shown was chosen so that the input current, I,, through the passive RC network is not significantly altered because of the feedback arrangement. It should be noted that if all capacitors in an RC network, for example, are grounded, then one could possibly dispense with the differential amplifiers and feed directly into the summing (operational) amplifier. It should be noted also that the feedback gain constants are a function of the impedances of the operational amplifier and the gains of the differential amplifiers. It is to be noted that 2,, Z 2,, K K and the buffer amplifier can each be used to control the feedback gain constants, K, and K of the system of FIG. 20. 2,, Z and Z, are impedances. For simplicity, these impedances could be-made to be resistors and thus Z,=R Z,=R Z =R would result. R and R could be varied in value to select the desired gains, K and K or a variation of the differential amplifiers K and K could have the same effect. Several approaches are possible.

Thus, if one fabricated all elements except the impedances, Z of the operational amplifier as one integrated circuit module, then by adding a second module with appropriate impedances, 2,, (say resistive) to the operational amplifier leads one could achieve any desired gains and thereby achieve any pole-zero configuration for that network. Thus, with one fixed network and a range of resistive modules, one could exercise pole-zero control. This is particularly attractive for integrated circuit applications since the small physical dimensions of the circuits lend themselves very readily to a modular design such as this. Also, as was noted previously, the feedback gain constants are a function of the operational amplifier impedance ratios, Z,/Z,.

This is also very important since in the fabrication of integrated element values, it is much easier to control the ratio of element values than the absolute values of those elements. In brief, a number of significant applications for this approach exist. For example, if after a long term operation of the circuit, environmental conditions, (aging, temperature, radiation, etc.) cause a shift of the circuit characteristics, one could easi' ly remedy the situation by simply replacing the resistive module (composed of the ratios, Z,/Z,, for all i) by another resistive module of an appropriate set of values.

Although the invention has been described with respect to preferred embodiments it is not to be so limited as changes and modifications may be made therein which are within the full and intended scope and as defined by the appended claims.

I claim as my invention:

1. The method of controlling the zeros of any passive driving-point immittance having at least one input port by applying state variable feedback to said one input port to control the zeros as a function of the feedback comprising synthesizing RL, RC, LC, RLC and nonpositive real driving point impedances from an active RC network comprising summing the voltages across the capacitors of said RC network and applying the results as feedback to said one input port of said network, and comprising summing the voltages across the capacitors of said RC network and applying the results as feedback at a port other than said one input port for pole control.

2. The method of claim 1 wherein the capacitor voltages are multiplied by gain constants before summing them and the gain constants vary as a function of the network to be synthesized.

3. The method of claim 2 comprising determining the network function of the desired network and selecting the gain constants based on the network function.

4. The method of claim 3 wherein the network function comprises numerator and denominator polynomials of the driving point impedance and the feedback factors of said one input port appear only in the numerator polynomial.

5. The method of claim 7 wherein the feedback factors at the port other than the input port is determined by both the numerator and denominator polynomials.

6. Apparatus for causing a resistor-capacitor network to respond as a network containing inductance comprising means for detecting the voltages across the capacitors in the network and means for combining the voltages across the capacitors and feeding the results into the network at their input port and wherein the means for combining the voltages across the capacitors modifies the amplitude of each voltage to obtain the desired network.

7. Apparatus according to claim 6 comprising means receiving the voltages across the capacitors and combining them to obtain a feedback signal and feeding the results into the net work at a port other than the input port.

8. The method of controlling the zeros of any passive driving-point immittance having at least one input port by applying state variable feedback to said one input port to control the zeros as a function of the feedback comprising synthesizing a desired network from a different: network comprising summing the currents in inductive elements and supplying them as feedback to the input port of said different network and comprising summing the currents in inductive elements and supplying them as feedback to ports other than said input port of the different networks.

9. The method of controlling the zeros of any passive driving-point immittance having at least one input port by applying state variable feedback to said one input port to control the zeros as a function of the feedback and comprising differentiating the state-variables and applying the differentiated signal to the input ports.

10. The method of claim 9 wherein the differentiated signal is applied to ports other than the input ports.

UNITED STATES PATENT OFFICE CERTIFICATE CF CRECTION Patent N 3 ,643,l84 Dated February 15, 1972 Inventor(s) John R. D'Alessandro It is certified that error appears in the above-identified patent and that said Letters Patent are hereby corrected as shown below:

Column 2, line 37, "network" should read work Column 4, line '55, "(5a) should read (s) Column 5, line 30, after "R insert line 31, after insert K same line "C should read C same line, after insert K line 54, in formula (13) after Z insert 2- Column 7, line 61, "design" should read designer Column 8, line 61, "Z /Z should read Z /Z Signed and sealed this 12th day of February 1974.

(SEAL) Attest:

EDWARD M.FLETCHER,JR. Cm MARSHALL DANN Attesting Officer Commissioner of Patents FORM po'wso uscoMM-Dc 60376-P69 U,S. GOVERNMENT PRINTING OFFICE: I969 O-366-334,

UNITED STATES PATENT OFFICE QERTIFICATE OF CDRRECTION Patent No. 3,6L 3, l8); Dated February 15, 1972 Inventor(s) IQhnR. D'Alessandro It is certified that error appears in the above-identified patent and that said Letters Patent arehereby corrected as shown below:

Column 9, line 26, replace "7" with --4--.

Signed and sealed this 21st day of Ma 1910+.

(SLEAL) Attest:

EDWARD I" ..I*"LL ET'JHBR,JR. C REESE- ALL DAM-K Attesting; Officer Commissionerof Patents 

1. The method of controlling the zeros of any passive drivingpoint immittance having at least one input port by applying state variable feedback to said one input port to control the zeros as a function of the feedback comprising synthesizing RL, RC, LC, RLC and nonpositive real driving point impedances from an active RC network comprising summing the voltages across the capacitors of said RC network and applying the results as feedback to said one input port of said network, and comprising summing the voltages across the capacitors of said RC network and applying the results as feedback at a port other than said one input port for pole control.
 2. The method of claim 1 wherein the capacitor voltages are multiplied by gain constants before summing them and the gain constants vary as a function of the network to be synthesized.
 3. The method of claim 2 comprising determining the network function of the desired network and selecting the gain constants based on the network function.
 4. The method of claim 3 wherein the network function comprises numerator and denominator polynomials of the driving point impedance and the feedback factors of said one input port appear only in the numerator polynomial.
 5. The method of claim 7 wherein the feedback factors at the port other than the input port is determined by both the numerator and denominator polynomials.
 6. Apparatus for causing a resistor-capacitor network to respond as a network containing inductance comprising means for detecting the voltages across the capacitors in the network and means for combining the voltages across the capacitors and feeding the results into the network at their input port and wherein the means for combining the voltages across the capacitors modifies the amplitude of each voltage to obtain the desired network.
 7. Apparatus according to claim 6 comprising means receiving the voltages across the capacitors and combining them to obtain a feedback signal and feeding the results into the network at a port other than the input port.
 8. The method of controlling the zeros of any passive driving-point immittance having at least one input port by applying state variable feedback to said one input port to control the zeros as a function of the feedback comprising synthesizing a desired network from a different network comprising summing thE currents in inductive elements and supplying them as feedback to the input port of said different network and comprising summing the currents in inductive elements and supplying them as feedback to ports other than said input port of the different networks.
 9. The method of controlling the zeros of any passive driving-point immittance having at least one input port by applying state variable feedback to said one input port to control the zeros as a function of the feedback and comprising differentiating the state-variables and applying the differentiated signal to the input ports.
 10. The method of claim 9 wherein the differentiated signal is applied to ports other than the input ports. 